Grand antiprism

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Grand antiprism
Grand antiprism.png
(Schlegel diagram wireframe)
Type Uniform 4-polytope
Uniform index47
Cells100+200 (3.3.3) Tetrahedron.png
20 (3.3.3.5) Pentagonal antiprism.png
Faces20 {5}
700 {3}
Edges500
Vertices100
Vertex figure Sphenocorona
Grand antiprism verf.png
Symmetry group Ionic diminished Coxeter group [[10,2<sup>+</sup>,10]] of order 400
Properties convex
Grand antiprism net.png
A net showing two disjoint rings of 10 antiprisms. 200 tetrahedra (yellow) are in face contact with the antiprisms and 100 tetrahedra (red) contact only other tetrahedra.

In geometry, the grand antiprism or pentagonal double antiprismoid is a uniform 4-polytope (4-dimensional uniform polytope) bounded by 320 cells: 20 pentagonal antiprisms, and 300 tetrahedra. It is an anomalous, non-Wythoffian uniform 4-polytope, discovered in 1965 by Conway and Guy. [1] [2] Topologically, under its highest symmetry, the pentagonal antiprisms have D5d symmetry and there are two types of tetrahedra, one with S4 symmetry and one with Cs symmetry.

Contents

Alternate names

Structure

20 stacked pentagonal antiprisms occur in two disjoint rings of 10 antiprisms each. The antiprisms in each ring are joined to each other via their pentagonal faces. The two rings are mutually perpendicular, in a structure similar to a duoprism.

The 300 tetrahedra join the two rings to each other, and are laid out in a 2-dimensional arrangement topologically equivalent to the 2-torus and the ridge of the duocylinder. These can be further divided into three sets. 100 face mate to one ring, 100 face mate to the other ring, and 100 are centered at the exact midpoint of the duocylinder and edge mate to both rings. This latter set forms a flat torus and can be "unrolled" into a flat 10×10 square array of tetrahedra that meet only at their edges and vertices. See figure below.

100 tetrahedra in a 10x10 array forming the Clifford torus boundary in the 600-cell and grand antiprism. 100 tets.jpg
100 tetrahedra in a 10×10 array forming the Clifford torus boundary in the 600-cell and grand antiprism.

In addition the 300 tetrahedra can be partitioned into 10 disjoint Boerdijk–Coxeter helices of 30 cells each that close back on each other. The two pentagonal antiprism tubes, plus the 10 BC helices, form an irregular discrete Hopf fibration of the grand antiprism that Hopf maps to the faces of a pentagonal antiprism. The two tubes map to the two pentagonal faces and the 10 BC helices map to the 10 triangular faces.

The structure of the grand antiprism is analogous to that of the 3-dimensional antiprisms. However, the grand antiprism is the only convex uniform analogue of the antiprism in 4 dimensions (although the 16-cell may be regarded as a regular analogue of the digonal antiprism ). The only nonconvex uniform 4-dimensional antiprism analogue uses pentagrammic crossed-antiprisms instead of pentagonal antiprisms, and is called the pentagrammic double antiprismoid .

Vertex figure

The vertex figure of the grand antiprism is a sphenocorona or dissected regular icosahedron: a regular icosahedron with two adjacent vertices removed. In their place 8 triangles are replaced by a pair of trapezoids, edge lengths φ, 1, 1, 1 (where φ is the golden ratio), joined together along their edge of length φ, to give a tetradecahedron whose faces are the 2 trapezoids and the 12 remaining equilateral triangles.

Tetrahedron vertfig.png
12 (3.3.3)
Pentagonal antiprism vertfig.png
2 (3.3.3.5)
Dissected regular icosahedron.png
Dissected regular icosahedron

Construction

The regular 600-cell can be decomposed with the symmetry of the grand antiprism, with each of the 20 blue pentagonal antiprisms being divided into 15 regular tetrahedra. 600-cell grand antiprism net.png
The regular 600-cell can be decomposed with the symmetry of the grand antiprism, with each of the 20 blue pentagonal antiprisms being divided into 15 regular tetrahedra.

The grand antiprism can be constructed by diminishing the 600-cell: subtracting 20 pyramids whose bases are three-dimensional pentagonal antiprisms. Conversely, the two rings of pentagonal antiprisms in the grand antiprism may be triangulated by 10 tetrahedra joined to the triangular faces of each antiprism, and a circle of 5 tetrahedra between every pair of antiprisms, joining the 10 tetrahedra of each, yielding 150 tetrahedra per ring. These combined with the 300 tetrahedra that join the two rings together yield the 600 tetrahedra of the 600-cell.

This diminishing may be realized by removing two rings of 10 vertices from the 600-cell, each lying in mutually orthogonal planes. Each ring of removed vertices creates a stack of pentagonal antiprisms on the convex hull. This relationship is analogous to how a pentagonal antiprism can be constructed from an icosahedron by removing two opposite vertices, thereby removing 5 triangles from the opposite 'poles' of the icosahedron, leaving the 10 equatorial triangles and two pentagons on the top and bottom.

(The snub 24-cell can also be constructed by another diminishing of the 600-cell, removing 24 icosahedral pyramids. Equivalently, this may be realized as taking the convex hull of the vertices remaining after 24 vertices, corresponding to those of an inscribed 24-cell, are removed from the 600-cell.)

Alternatively, it can also be constructed from the decagonal ditetragoltriate (the convex hull of two perpendicular nonuniform 10-10 duoprisms where the ratio of the two decagons are in the golden ratio) via an alternation process. The decagonal prisms alternate into pentagonal antiprisms, the rectangular trapezoprisms alternate into tetrahedra with two new regular tetrahedra (representing a non-corealmic triangular bipyramid) created at the deleted vertices. This is the only uniform solution for the p-gonal double antiprismoids alongside its conjugate, the pentagrammic double antiprismoid from the decagrammic ditetragoltriate.

Orthogonal projections
600-cellGrand antiprism
H4 Coxeter plane
600-cell graph H4.svg Grand antiprism ortho-30-gon.png
20-gonal
600-cell t0 p20.svg Grand antiprism 20-gonal orthogonal projection.png
H3 Coxeter plane (slight offset)
Grand antiprism 600-cell H3.png

Projections

These are two perspective projections, projecting the polytope into a hypersphere, and applying a stereographic projection into 3-space.

Stereographic grand antiprism.png
Wireframe, viewed down one of the pentagonal antiprism columns.
Stereographic grand antiprism faces.png
with transparent triangular faces
Ortho solid 963-uniform polychoron grand antiprism.png
Orthographic projection
Centered on hyperplane of an antiprism in one of the two rings.
GrandAntiPrism-3DOrtho-30.png
3D orthographic projection
of 100 of 120 600-cell vertices and 500 edges {488 of 1/2 (3-Sqrt[5]) and 12 of 2/(3+Sqrt[5])}.

See also

Notes

  1. J.H. Conway and M.J.T. Guy: Four-Dimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965. (Michael Guy is son of Richard K. Guy)
  2. Conway, 2008, p.402-403 The Grand Antiprism
  3. Klitzing, Richard. "4D convex polychora Grand antiprism".

Related Research Articles

600-cell Four-dimensional analog of the icosahedron

In geometry, the 600-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,5}. It is also known as the C600, hexacosichoron and hexacosihedroid. It is also called a tetraplex (abbreviated from "tetrahedral complex") and a polytetrahedron, being bounded by tetrahedral cells.

Uniform 4-polytope

In geometry, a uniform 4-polytope is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons.

Duoprism

In geometry of 4 dimensions or higher, a duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher. The Cartesian product of an n-polytope and an m-polytope is an (n+m)-polytope, where n and m are 2 (polygon) or higher.

Rectified 600-cell

In geometry, the rectified 600-cell or rectified hexacosichoron is a convex uniform 4-polytope composed of 600 regular octahedra and 120 icosahedra cells. Each edge has two octahedra and one icosahedron. Each vertex has five octahedra and two icosahedra. In total it has 3600 triangle faces, 3600 edges, and 720 vertices.

Great icosahedron

In geometry, the great icosahedron is one of four Kepler-Poinsot polyhedra, with Schläfli symbol {3,52} and Coxeter-Dynkin diagram of . It is composed of 20 intersecting triangular faces, having five triangles meeting at each vertex in a pentagrammic sequence.

Snub 24-cell

In geometry, the snub 24-cell or snub disicositetrachoron is a convex uniform 4-polytope composed of 120 regular tetrahedral and 24 icosahedral cells. Five tetrahedra and three icosahedra meet at each vertex. In total it has 480 triangular faces, 432 edges, and 96 vertices. One can build it from the 600-cell by diminishing a select subset of icosahedral pyramids and leaving only their icosahedral bases, thereby removing 480 tetrahedra and replacing them with 24 icosahedra.

Runcinated 24-cells

In four-dimensional geometry, a runcinated 24-cell is a convex uniform 4-polytope, being a runcination of the regular 24-cell.

Runcinated 120-cells

In four-dimensional geometry, a runcinated 120-cell is a convex uniform 4-polytope, being a runcination of the regular 120-cell.

Uniform polytope

A uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons.

Uniform 5-polytope

In geometry, a uniform 5-polytope is a five-dimensional uniform polytope. By definition, a uniform 5-polytope is vertex-transitive and constructed from uniform 4-polytope facets.

Great duoantiprism

The great duoantiprism is the only uniform star-duoantiprism solution p=5, q=5/3, in 4-dimensional geometry. It has Schläfli symbol {5}⊗{5/3}, s{5}s{5/3} or ht0,1,2,3{5,2,5/3}, Coxeter diagram , constructed from 10 pentagonal antiprisms, 10 pentagrammic crossed-antiprisms, and 50 tetrahedra.

5-5 duoprism

In geometry of 4 dimensions, a 5-5 duoprism or pentagonal duoprism is a polygonal duoprism, a 4-polytope resulting from the Cartesian product of two pentagons.

8-8 duoprism

In geometry of 4 dimensions, a 8-8 duoprism or octagonal duoprism is a polygonal duoprism, a 4-polytope resulting from the Cartesian product of two octagons.

6-6 duoprism

In geometry of 4 dimensions, a 6-6 duoprism or hexagonal duoprism is a polygonal duoprism, a 4-polytope resulting from the Cartesian product of two hexagons.

10-10 duoprism

In geometry of 4 dimensions, a 10-10 duoprism or decagonal duoprism is a polygonal duoprism, a 4-polytope resulting from the Cartesian product of two decagons.

4-6 duoprism

In geometry of 4 dimensions, a 4-6 duoprism, a duoprism and 4-polytope resulting from the Cartesian product of a square and a hexagon.

4-8 duoprism

In geometry of 4 dimensions, a 4-8 duoprism, a duoprism and 4-polytope resulting from the Cartesian product of a square and an octagon.

In geometry of 4 dimensions, a 6-8 duoprism, a duoprism and 4-polytope resulting from the Cartesian product of a hexagon and an octagon.

References

Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds